1. Technical Field:
The present invention relates to image restoration, and more particularly to a regularized Bayesian framework for image restoration.
2. Discussion of Related Art:
Bayesian frameworks have been used in different applications including image processing (e.g., image restoration, stereo/motion estimation or segmentation), machine learning (e.g., hidden Markov model or graphical models), etc. In the context of multiple image restoration, conventional Bayesian methods are sensitive to model errors and cannot guarantee valid results satisfying the underlying prior knowledge, e.g., independent noise property.
In image restoration numerous methods (e.g. Wiener filter, steerable filters, Wavelet) have been proposed to improve image quality and reduce imaging noise. Bayesian restoration schemes have been explored to find the maximum a posteriori (MAP) estimation of a true signal based on statistical noise/signal models (i.e., generative models). Desirable results have been achieved when accurate models can be trained in advance. MAP can be formulated as:
                              S          ^                =                              arg            ⁢                                                  ⁢                                          max                S                            ⁢                              P                ⁡                                  (                                      S                    |                    I                                    )                                                              =                      arg            ⁢                                                  ⁢                                          max                S                            ⁢                                                P                  ⁡                                      (                                          I                      |                      S                                        )                                                  ⁢                                  P                  ⁡                                      (                    S                    )                                                                                                          (        1        )            where I is an observed noisy image and S is the true image to be recovered. P(S) models the expected structures in the true image (e.g., smooth surfaces, step edges or corners). P(I|S) is the conditional distribution of the observed image I given the true image S. MAP models how the observed image is generated, and can include point spread functions and noise models. The MAP estimation is obtained by finding Ŝ that has the maximum probability: P(Ŝ|I).
Multiple images can be obtained in some cases to further improve imaging quality (e.g., ultrasound spatial/frequency compounding or multi-spectral remote sensing). A Bayesian framework can be extended to multiple images when the images are corrupted by independent noise.
Methods for improving the signal noise models attempt to discriminate the signal and noise (e.g., MRF based edge modeling, Wavelet or AQua model). Optimization procedures have also been proposed. For multi-image restoration, the correlations between the signal in different images are exploited. For example, a coupled edge modeling on multi-image has been proposed to achieve better edge detection and better edge preservation during noise reduction. In real world applications, it can be difficult to obtain accurate prior models. For example, the ultrasound speckle noise is non-stationary and changes according to ultrasound attenuation and the sub-resolution scatterers in the tissue. Various types of structures (e.g., corners, edges or surfaces) are also hard to model accurately.
Further, over-simplified assumptions (e.g., the noise being independent of the signal) may need to be made to allow tractable solutions. Under these difficult conditions, the conventional Bayesian framework cannot provide robust results and some of the underlying prior knowledge/constraints may even be violated. One important prior in multi-image restoration is that the multiple images are often corrupted by independent noise, which is the very basis for many Bayesian restoration methods to factorize the joint probabilities and hence allows tractable solutions. However, this prior is also under-utilized and often violated one when there are modeling errors. An inaccurate weighting between noise and signal models results in significantly correlated noise. Such is violations of the prior constraints indicate restoration errors and sensitivity to inaccurate models of the conventional Bayesian framework. In machine learning, Support Vector Machine (SVM) has been proposed to replace the generative model based Bayesian methods for better generalization. In image restoration, however, generative models of signal and noise have been extensively studied and are important for discriminating noise and signal. It is important not to forfeit those generative models.
Therefore, a need exists for a Bayesian system and method having improved generalization and enforced validity.